Source-linked AI summary
Interval-valued neutrosophic soft sets and its decision making
Irfan Deli
TL;DR
The paper addresses uncertainty modeling by defining interval-valued neutrosophic soft sets. It develops their operations and properties, then uses level soft sets for adjustable decision making and illustrates the approach with examples.
Problem
The paper addresses the need to model uncertain, imprecise, incomplete, and inconsistent information while making neutrosophic sets more applicable to real applications.
Method
The paper combines interval-valued neutrosophic sets with soft sets, defines operations and properties, and develops an adjustable level-soft-set decision-making approach.
Results
The paper establishes properties including smallest-union, largest-intersection, De Morgan, and distributive-lattice results for interval-valued neutrosophic soft sets.
Takeaways & Limitations
Level soft sets bridge interval-valued neutrosophic soft sets and crisp soft sets for decision making, and the proposed approach is reported feasible for some such problems.
Abstract
from arXiv · showhide
In this paper, the notion of the interval valued neutrosophic soft sets ($ivn-$soft sets) is defined which is a combination of an interval valued neutrosophic sets \cite{wan-05} and a soft sets \cite{mol-99}. Our $ivn-$soft sets generalizes the concept of the soft set, fuzzy soft set, interval valued fuzzy soft set, intuitionistic fuzzy soft set, interval valued intuitionistic fuzzy soft set and neutrosophic soft set. Then, we introduce some definitions and operations on $ivn-$soft sets sets. Some properties of $ivn-$soft sets which are connected to operations have been established. Also, the aim of this paper is to investigate the decision making based on $ivn-$soft sets by level soft sets. Therefore, we develop a decision making methods and then give a example to illustrate the developed approach.
1 Introduction
The paper motivates soft-set extensions for uncertainty and introduces interval neutrosophic soft sets as a broader model for uncertain, imprecise, incomplete, and inconsistent information.
- Classical mathematics may not successfully model uncertain data because uncertainty is complex and not clearly defined.
- Soft set theory models uncertainty from a parametrization perspective using classical sets.
- Soft sets have been applied across algebra, ontology, optimization, topology, data analysis, operations research, clustering, medical diagnosis, and decision making.
- Neutrosophic sets explicitly quantify indeterminacy and allow truth, indeterminacy, and falsity memberships to be independent.
- Interval neutrosophic soft sets generalize soft, fuzzy soft, interval-valued fuzzy soft, intuitionistic fuzzy soft, interval-valued intuitionistic fuzzy soft, and neutrosophic soft sets.
- The paper develops definitions, operations, related properties, and a level-soft-set decision-making approach for interval neutrosophic soft sets.
2 Preliminary
The preliminary section introduces neutrosophic, interval-valued neutrosophic, and soft-set concepts, together with their representations, examples, and operations.
- Neutrosophic sets characterize each object with truth-membership, indeterminacy-membership, and falsity-membership functions.
- There is no restriction requiring the sum of truth, indeterminacy, and falsity memberships to equal one; their upper-bound sum lies between 0 and 3.
- Interval-valued neutrosophic sets represent truth, indeterminacy, and falsity memberships as intervals within [0, 1].
- An interval-valued neutrosophic number is the triple of truth, indeterminacy, and falsity membership values for an object.
- A soft set maps selected parameters X ⊆ E to subsets of a universe U through an approximate function.
- Soft-set operations introduced here include complement, union, and intersection, with examples and tabular representations.
3 Interval-valued neutrosophic soft sets
This section defines interval-valued neutrosophic soft sets as a combination of interval-valued neutrosophic sets and soft sets, then develops their operations and associated properties. It introduces basic examples, set relations, algebraic operations, and lattice structure.
- Definition and representation: Interval-valued neutrosophic soft sets combine interval-valued neutrosophic sets with soft sets and generalize several established soft-set variants.The paper presents them as generalizations of soft, fuzzy soft, interval-valued fuzzy soft, intuitionistic fuzzy soft, interval-valued intuitionistic fuzzy soft, and neutrosophic soft sets.
- Definition and representation: An ivn-soft set uses a set-valued approximate function that maps parameters to interval-valued neutrosophic sets over the universe.The approximate function and its parameter-specific values provide the representation used throughout the definitions and examples.
- Definitions and operations: The section defines empty, universal, subset, equality, complement, union, intersection, OR, AND, difference, addition, scalar multiplication, and scalar division operations.Examples and tabular representations illustrate several of these constructions for ivn-soft sets and their derived forms.
- Set relations: The union of two ivn-soft sets is the smallest ivn-soft set containing both, while their intersection is the largest ivn-soft set containing both.These extremal characterizations are stated separately for union and intersection.
- Algebraic properties: The power set of all ivn-soft sets over a universe forms a distributive lattice under ivn-soft intersection and union.The stated proof invokes idempotency, commutativity, associativity, and distributivity.
4 ivn−soft set based decision making
The section extends level-soft-set decision making to interval-valued neutrosophic soft sets by defining threshold constructions, level soft sets, and an adjustable decision procedure. Examples illustrate threshold-based selection and the resulting choice process.
- Decision-making framework: The approach extends level-soft-set decision making from interval-valued intuitionistic fuzzy soft sets to ivn-soft sets.The paper explicitly motivates this extension from prior level-soft-set definitions and applies it to ivn-soft-set decision problems.
- Level soft sets: An (α, β, γ)-level soft set converts an ivn-soft set into a crisp soft set using truth, indeterminacy, and falsity thresholds.Truth-membership uses a least threshold, while indeterminacy-membership and falsity-membership use greatest thresholds.
- Illustrative example: For a worked example, the ([0.3, 0.4], [0.3, 0.5], [0.1, 0.2])-level soft set contains x1 paired with {u1} and x4 paired with {u1, u2}.This example demonstrates how specified interval thresholds produce a crisp level soft set.
- Level soft sets: Different thresholds can be assigned to different parameters to accommodate parameter-specific decision-maker requirements.The paper replaces constant thresholds with threshold functions when parameters require different truth-, indeterminacy-, or falsity-membership criteria.
- Threshold rules: Average, max-min-min, and min-min-min thresholds summarize interval-valued neutrosophic information for corresponding level-based decision rules.The avg rule uses an average threshold, while Mmm and mmm rules use max-min-min and min-min-min threshold constructions, respectively.
- Decision algorithm: The decision algorithm inputs a threshold, computes the corresponding level soft set, tabulates it, and computes choice values ci for objects ui.If multiple objects share the relevant choice value, the procedure permits selecting any one of them.
5 Conclusion
The paper defines ivn-soft sets, develops their operations and related properties, and proposes an adjustable level-soft-set approach for decision making. Concrete examples support the approach's feasibility for uncertainty-related decision problems.
- Conclusion: The paper defines ivn-soft sets by combining interval-valued neutrosophic sets with soft sets.The proposed structure is presented as a new soft-set form for representing interval-valued neutrosophic information.
- Conclusion: It introduces ivn-soft-set definitions and operations, establishes properties connected with those operations, and develops an adjustable level-soft-set decision approach.The approach is illustrated through concrete examples and is described as feasible for some decision-making problems involving ivn-soft sets.
- Conclusion: The proposed method is applicable to decision problems involving uncertainty in fields such as computer science and game theory.The conclusion states this application scope without claiming universal effectiveness across all such problems.